In previous grades, students learned strategies for multiplication and division, developed understanding of structure of the place value system, and applied understanding of fractions to addition and subtraction with like denominators. Students gained understanding that geometric figures can be analyzed and classified based on their properties. The Grade 5 year in this scope and sequence begins with understanding volume to engage student interest and allow them to apply their understanding of operations. Students extend their understanding of place value to decimals and use the four operations with decimals. Place value is an area of mastery for fifth grade. Fluency of addition and subtraction of fractions is developed throughout the year. Students also develop understanding of multiplication and division of fractions. This scope and sequence assumes 160 days for instruction, divided among 15 units. The units are sequenced in a way that we believe best develops and connects the mathematical content described in the Common Core State Standards for Mathematics; however, the order of the standards included in any unit does not imply a sequence of content within that unit. Some standards may be revisited several times during the course; others may be only partially addressed in different units, depending on the mathematical focus of the unit. Throughout Grade 5, students should continue to develop proficiency with the Common Core's eight Standards for : These practices should become the natural way in which students come to understand and to do mathematics. While, depending on the content to be understood or on the problem to be solved, any practice might be brought to bear, some practices may prove more useful than others. Opportunities for highlighting certain practices are indicated in different units of study in this sample scope and sequence, but this highlighting should not be interpreted to mean that other practices should be neglected in those units. This scope and sequence reflects our current thinking related to the intent of the CCSS for Mathematics, but it is an evolving document. We expect to make refinements to this scope and sequence in the coming months in response to new learnings about the standards. In planning your district's instructional program, you should be prepared to have similar flexibility in implementing your district's own scope and sequence for the next 2 to 3 years, as you transition from your state's current standards to full implementation of the CCSS for Mathematics. at The University of Texas at Austin 10/31/11 1
Understanding volume 5.MD.3.a.b (Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a. A cube with side length 1 unit, called a unit cube, is said to have one cubic unit of volume, and can be used to measure volume. b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units. 5.MD.4 (Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.) 14 This is a very student- engaging new concept that builds on the understanding of area from 4th grade. The computations are very simple from the beginning; all students have access. at The University of Texas at Austin 10/31/11 2
Place value of 12 decimals 5.NBT.1 (Recognize that in a multi- digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.) 5.NBT.2 (Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole- number exponents to denote powers of 10.) 5.NBT.3.a.b (Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using base- ten numerals, number names, and expanded form, e.g., 347.392 = 3 x 100 + 4 x 10 + 7 x 1 + 3 x (1/10) + 9 x (1/100) + 2 x (1/1000). b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.) 5.NBT.4 (Use place value understanding to round decimals to any place.) Addition and subtraction of decimals 5.NBT.7 (Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.) 5.OA.1 (Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.) 14 at The University of Texas at Austin 10/31/11 3
Using and 5.NBT.5 (Fluently multiply multi- digit whole numbers using the standard algorithm.) 5.NBT.7 applying 5.MD.1 (Convert among different- sized standard measurement units within a given multiplication measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving with whole multi- step, real world problems.) numbers and decimals 14 This allows students to practice the standard algorithm. The new part is multliplying decimals to hundredths. Classifying 2- dimensional figures 5.G.3 (Understand that attributes belonging to a category of two- dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.) 5.G.4 (Classify two- dimensional figures in a hierarchy based on properties.) 10 at The University of Texas at Austin 10/31/11 4
Relating volume 7 to multiplication and addition 5.MD.5.a.b.c (Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. a. Find the volume of a right rectangular prism with whole- number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole- number products as volumes, e.g., to represent the associative property of multiplication. b. Apply the formulas V = l x w x h and V = b x h for rectangular prisms to find volumes of right rectangular prisms with whole- number edge lengths in the context of solving real world and mathematical problems. c. Recognize volume as additive. Find volumes of solid figures composed of two non- overlapping right rectangular prisms by adding the volumes of the non- overlapping parts, applying this technique to solve real world problems. Dividing whole numbers and decimals 5.NBT.6 (Find whole- number quotients of whole numbers with up to four- digit dividends and two- digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.) 5.NBT.7 5.MD.1 12 at The University of Texas at Austin 10/31/11 5
Seeing fractions 5 as division 5.NF.3 (Interpret a fraction as division of the numerator by the denominator (a/b = a b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50- pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?) at The University of Texas at Austin 10/31/11 6
Strategies for 12 addition and subtraction of fractions with unlike denominators 5NF.1 (Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)) 5NF.2 (Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.) Using ordered pairs 5.G.1 (Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x- axis and x- coordinate, y- axis and y- coordinate).) 5.G.2 (Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.) 5.OA.3 (Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule Add 3 and the starting number 0, and given the rule Add 6 and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.) 11 at The University of Texas at Austin 10/31/11 7
Multiplying 12 fractions Problem solving with decimals and whole numbers 5.NF.4.a.b (Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) x q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a x q b. For example, use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation. Do the same with (2/3) x (4/5) = 8/15. (In general, (a/b) x (c/d) = ac/bd.) b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.) 5.NF.5.a.b (Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (nxa)/(nxb) to the effect of multiplying a/b by 1. 5.NF.6 (Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.) 5.OA.1 (Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.) 5.OA.2 (Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating For example, express the calculation add 8 and 7, then multiply by 2 as 2 x (8 + 7). Recognize that 3 x (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.) 5.NBT.7 10 at The University of Texas at Austin 10/31/11 8
Dividing 10 fractions Representing and interpreting data with fractions 5.NF.7.a.b.c (Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. 1 a. Interpret division of a unit fraction by a non- zero whole number, and compute such quotients. For example, create a story context for (1/3) 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) 4 = 1/12 because (1/12) 4 = 1/3. b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 (1/5) = 20 because 20 (1/5) = 4. c. Solve real world problems involving division of unit fractions by non- zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3- cup servings are in 2 cups of raisins?) 1 Students able to multiply fractions in general can develop strategies to divide frac tions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade. 5MD.2 (Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.) 7 at The University of Texas at Austin 10/31/11 9
Problem solving 13 with fractions 5.NF.2 (Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.) 5.NF.6 5.NF.7.c at The University of Texas at Austin 10/31/11 10